On the roles of “stability” and “convergence” in semidiscrete projection methods for initial-value problems

Abstract:

Consider the initial value problem (1.1) \[ \frac {d}{{dt}}u(t) = Au(t) + f(t),\quad t > 0,\] (1.2) \[ u(0) = {u_0},\] where A is a linear operator taking $D(A) \subset X$ into X, where X is a Banach space. Consider also semidiscrete numerical methods of the form: find ${U_N}(t):[0,T] \to {X_N}$ such that $(1.1\prime )$ \[ \frac {{d{U_N}}}{{dt}} = {A_N}{U_N} + {P_N}f,\] $(1.2\prime )$ \[ {U_N}(0) = U_N^0 \in {X_N},\] where ${X_N}$ is a finite dimensional subspace and ${P_N}$ is a projector onto ${X_N}$. The study of such numerical methods may be related to the approximation of semigroups and Laplace transform methods making use of the resolvent operators ${(A - \lambda I)^{ - 1}},{({A_N} - \lambda {I_N})^{ - 1}}$. The basic results require stability or weak stability and give convergence rates of the same order as in the steady state problems.